Division Algorithm in F[x]

Abstract Algebra · Axiom Academy

EXAMPLE Division Algorithm in F[x] Perform polynomial long division step-by-step in ℚ[x] and verify the division algorithm. Excellent work! You've successfully applied the Division Algorithm in F[x]. Here's what we learned: Division Algorithm Statement: For with , there exist unique polynomials q(x) and r(x) such that , where either r(x) = 0 or . Long Division Process: Divide leading terms, multiply the entire divisor, subtract, and repeat until the remainder has degree less than the divisor. Stopping Condition: The process terminates when ( ) < ( ) . In our case, (2x)=1 < (x^2 - x + 1)=2 . Verification: Always check that . We verified: (x^2 - x + 1)(x + 1) + 2x = x^3 + 2x + 1 ✓ Uniqueness: The quotient and remainder are unique for any given dividend and divisor in F[x] . The Division Algorithm is fundamental in Abstract Algebra, leading to important concepts like greatest common divisors, irreducibility, and field extensions. This same structure works in any polynomial ring F[x] where F is a field!

This is the written version of the interactive lesson above. See the full Abstract Algebra course.